Euler’s phi function

The number of members of this sequence, which are relatively prime to the number , we denote as . In this way we obtain a function defined with . We call this function the Euler’s phi function and it is very important in the number theory.

For instance, find . We observe the sequence:

Numbers that are relatively prime to are and , and there are , so .

From the definition follows that for every prime number ,

is valid.

The question now is how would we determine for ”large” . That help us the multiplicative property of the Euler’s phi function, given in the following theorem.

Theorem 1. If and are relatively prime numbers, then

Example 1. Determine .

Solution.

We write the number as the product of two relatively prime factors: . By the formula above

is valid. Since and , it follows

More effective formula to determine the number is given in the following theorem.

Theorem 2. If is a prime factor decomposition of the natural number , then